The infimum of the energy of unit vector fields on odd-dimensional spheres
Let \(M\) be an oriented, closed, \(n\)-dimensional Riemannian manifold and \(T^1M\) be the unit tangent bundle of \(M\) considered as a closed Riemannian manifold with the Sasaki metric. The energy of a unit smooth vector field \(v\) on \(M\) is \(E(v)= {1\over 2}\int_M(n+\|\nabla\|^2)dM\), where \(dM\) is the volume form on \(M\). The authors construct a one-parameter family of unit smooth vector fields globally defined on the sphere \(\mathbb{S}^{2k +1}\) for \(k\geq 2\), with energy converging to the energy of the unit radial vector field defined on the complement of two antipodal points. The infimum of the energy of these vector fields is \(({2k+1\over 2}+ {k\over 2k-1}) \text{vol} (\mathbb{S}^{2k+1})\).
scientific article; zbMATH DE number 1925034